First thing that is worth mentioning is fact that there are several models of using the cloud in general. Everybody knows and understands what the public cloud is - everybody can use resources as much as they want from public share. Even individuals can buy some resources.
But not everybody is happy in this model - there are some companies that doesn't want to share physically their data with other companies. Of course the data stored in data centers are much safer than every other place, but it's very difficult to convince some people to this new model of computing.
Anyway, we have in major two types of clouds: public and private. In public cloud the resources are being shared between everybody and in private cloud we don't share any physically machine or data with nobody.
But there are some situations when this is not working. Because of financial limitations private clouds are in usual smaller than public massive cloud solutions. So sometimes when they need more compute power they're forced to use public cloud to cope with some unpredictable workloads. It's called "cloudbursting" - really fancy name that in my opinion hides really good fears from using the public cloud.
So cloudbursting is using public cloud to outsource some computing out of our own cloud solution. It shouldn't be used too much, because that would mean that our private cloud isn't powerful enough and / or we're loosing too much time for developing cloudbursting solutions which can be nontrivial. So if this bursts are happening too often it is a first signal that something is wrong with architecture of our services. It should happen only on spikes of load.
Cloudbursting is a first step to moving whole company to the public cloud.